Electronic Journal of Differential Equations (Feb 2014)
Asymptotic behavior of singular solutions to semilinear fractional elliptic equations
Abstract
In this article we study the asymptotic behavior of positive singular solutions to the equation $$ (-\Delta)^{\alpha} u+u^p=0\quad\text{in } \Omega\setminus\{0\}, $$ subject to the conditions $u=0$ in $\Omega^c$ and $\lim_{x\to0}u(x)=\infty$, where $p\geq1$, $\Omega$ is an open bounded regular domain in $\mathbb{R}^N$ ($N\ge2$) containing the origin, and $(-\Delta)^\alpha$ with $\alpha\in(0,1)$ denotes the fractional Laplacian. We show that the asymptotic behavior of positive singular solutions is controlled by a radially symmetric solution with $\Omega$ being a ball.